Procesamiento de Señales e Imagenes

Ingeniería Biomédica

Ph.D. Pablo Eduardo Caicedo Rodríguez

2025-09-07

Procesado de Señales e Imágenes Médicas - PSIM

Introduction

Biosignals

  • Bio - That came from a biological being
  • Signal - A signal is a function that conveys information about a physical phenomenon.
  • Biosignals - The search for information from living systems to know its health state

Introduction

Biosignals

The codification of biosignals into variations: * Electrical * Mechanical * Chemical * Thermal

Introduction

Taken from Semmlow et al

Introduction

Taken from Semmlow et al

Time line

Biomedical Signals:

  • 1791: Luigi Galvani discovers electrical signals in living tissues (frog legs)

  • 1830s: Carlo Matteucci studies electrical signals in the heart

  • 1887: Willem Einthoven invents the first electrocardiograph (ECG)

  • 1900s: James Mackenzie develops the first clinical ECG machine

Time line

Biomedical Signals:

  • 1920s: Electroencephalography (EEG) is developed by Hans Berger

  • 1930s: Electromyography (EMG) is developed by John Humphrey and others

  • 1940s: Development of the first commercial ECG machines

  • 1950s: Signal processing techniques are applied to biomedical signals

Time line

Biomedical Signals:

  • 1960s: Digital signal processing and computer analysis of biomedical signals emerge

  • 1970s: Biomedical signal processing becomes a recognized field

  • 1980s: Development of Holter monitoring (24-hour ECG)

  • 1990s: Advances in signal processing and machine learning applied to biomedical signals

Time line

Biomedical Signals:

  • 2000s: Development of wearable devices and mobile health (mHealth) technologies

  • 2010s: Emergence of big data analytics and cloud computing in biomedical signal processing

  • 2020s: Integration of artificial intelligence (AI) and machine learning (ML) in biomedical signal processing

Time line

Biomedical Images:

  • 1895: Wilhelm Roentgen discovers X-rays, leading to medical imaging

  • 1900s: X-ray technology improves with development of modern X-ray tubes

  • 1913: Albert Salomon develops mammography

  • 1920s: Ultrasound technology is developed by Karl Dussik and others

Time line

Biomedical Images:

  • 1930s: Nuclear medicine emerges with development of radioactive tracers

  • 1950s: Computed Tomography (CT) scans are developed by Godfrey Hounsfield and Allan McLeod Cormack

  • 1960s: Development of medical ultrasound imaging

  • 1970s: Magnetic Resonance Imaging (MRI) is developed by Richard Ernst and others

Time line

Biomedical Images:

  • 1980s: Digital image processing and analysis techniques are applied to biomedical images

  • 1990s: Advances in MRI and CT scan technology, including 3D imaging

  • 2000s: Development of functional MRI (fMRI), diffusion tensor imaging (DTI), and other advanced MRI techniques

  • 2010s: Emergence of artificial intelligence (AI) and machine learning in medical imaging

Time line

Additional Milestones:

  • 1950s: Development of medical electronics and instrumentation
  • 1960s: First medical imaging computers are developed
  • 1970s: Development of digital image processing and analysis software
  • 1980s: Emergence of medical imaging informatics and PACS (Picture Archiving and Communication Systems)
  • 1990s: Development of telemedicine and teleradiology
  • 2000s: Emergence of electronic health records (EHRs) and health information exchanges (HIEs)
  • 2010s: Development of personalized medicine and precision health initiatives

Part I: Probability and Statistics (Repaso)

Events, Sample Space, Experiments

Definition

An experiment is a physical procedure that produces some kind of result.

Definition

An event is a set of experiment’s possible results.

Consejo

A sample space is the set of ALL possibles results of an experiment.

Events, Sample Space, Experiments

data = np.genfromtxt("../../data/mitbih_train.csv", delimiter=",")
ecg1 = data[1, :-1]
time = np.array(range(0,len(ecg1)))/125
fig = plt.figure()
plt.plot(time, ecg1)
plt.xlabel("Time (s)")
plt.ylabel("Normalized ECG")
print("Maximun Value: "+ str(ecg1.max()))
Maximun Value: 1.0
print("Minimun Value: "+ str(ecg1.min()))
Minimun Value: 0.0
print(ecg1[np.random.choice(ecg1.shape[0], 1, replace=False)])
[0.]

Name: ECG Heartbeat Categorization Dataset.

URL: https://www.kaggle.com/datasets/shayanfazeli/heartbeat?resource=download

Probability Axioms

For the given events A and B that are in a sample space S:

Axioms

  • \(0 \leq P_r \left(A\right) \leq 1\)
  • \(P_r\left(S\right) = 1\)
  • If \(A \cap B = \emptyset\) then \(P_r\left(A \cup B \right) = P_r \left(A\right) + P_r \left(B\right)\)
  • If \(A \cap B \neq \emptyset\) then \(P_r\left(A \cup B \right) = P_r \left(A\right) + P_r \left(B\right) - P_r\left(A \cap B \right)\)
  • \(P_r\left(\bar{A}\right) = 1-P_r \left(A\right)\)
  • If \(A\subset B\) then \(P_r \left(A\right)\leq P_r \left(B\right)\)
  • \(P_r \left(A|B\right)=\frac{P_r \left(A\cap B\right)}{P_r \left(B\right)}\)

Random Variable

Definition

A random variable is a real valued function of the elements of a sample space, S . Given an experiment, E , with sample space, S, the random variable maps each possible outcome of E.

Definition

The probability mass function (PMF), \(P_X\left(x\right)\), of a random variable, X, is a function that assigns a probability to each possible value of the random variable, X.

Random Variable

Random Variables

Conditions

Discrete

\[\sum_{\chi \in X}P_X\left(\chi \right) = 1\]

Continuous

\[\int_{-\infty}^{\infty}P_X\left(\chi \right)d\chi = 1\]

Random Variables

Expected Values

Discrete

\[\mu = \sum_{\chi \in X}\chi P_X\left(\chi \right)\]

Continuous

\[\mu=\int_{-\infty}^{\infty}\chi P_X\left(\chi \right)d\chi\]

Random Variables

Variance

Discrete

\[\sigma^2 = \sum_{\chi \in X}\left(\chi - \mu \right)^2 P_X\left(\chi \right)\]

Continuous

\[\sigma^2 = \int_{-\infty}^{\infty}\left(\chi - \mu \right)^2 P_X\left(\chi \right)d\chi\]

PDF Estimation

counts01, bin_edges01 = np.histogram(ecg1, bins=10, density=True)
counts02, bin_edges02 = np.histogram(ecg1, bins=50, density=True)
counts03, bin_edges03 = np.histogram(ecg1, bins=100, density=True)
fig01=plt.figure()
plt.plot(bin_edges01[1:], counts01/sum(counts01), label="Estimation with 10 bins")
plt.plot(bin_edges02[1:], counts02/sum(counts02), label="Estimation with 50 bins")
plt.plot(bin_edges03[1:], counts03/sum(counts03), label="Estimation with 100 bins")
plt.legend()
plt.grid()
plt.xlabel("Normalised ECG Value")
plt.ylabel("Estimated PDF Value")
0.09001020772910533
0.02551116143316462